A Flexible New Technique for Camera Calibration

Transcription

1 A Flexible New Technique for Camera Calibration Zhengyou Zhang December 2, 1998 (updated on December 14, 1998) (updated on March 25, 1999) (updated on Aug. 1, 22; a typo in Appendix B) (updated on Aug. 13, 28; a typo in Section 3.3) (last updated on Dec. 5, 29; a typo in Section 2.4) Technical Report MSR-TR Citation: Z. Zhang, A flexible new technique for camera calibration, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11): , 2. Microsoft Research Microsoft Corporation One Microsoft Way Redmond, WA 9852 [email protected] zhang

2 A Flexible New Technique for Camera Calibration Zhengyou Zhang Microsoft Research, One Microsoft Way, Redmond, WA , USA zhang Contents 1 Motivations 2 2 Basic Equations Notation Homography between the model plane and its image Constraints on the intrinsic parameters Geometric Interpretation Solving Camera Calibration Closed-form solution Maximum likelihood estimation Dealing with radial distortion Summary Degenerate Configurations 8 5 Experimental Results Computer Simulations Real Data Sensitivity with Respect to Model Imprecision Random noise in the model points Systematic non-planarity of the model pattern Conclusion 17 A Estimation of the Homography Between the Model Plane and its Image 17 B Extraction of the Intrinsic Parameters from Matrix B 18 C Approximating a 3 3 matrix by a Rotation Matrix 18 D Camera Calibration Under Known Pure Translation 19 added on December 14, 1998 added on December 28, 1998; added results on systematic non-planarity on March 25, 1998 added on December 14, 1998, corrected (based on the comments from Andrew Zisserman) on January 7,

3 A Flexible New Technique for Camera Calibration Abstract We propose a flexible new technique to easily calibrate a camera. It is well suited for use without specialized knowledge of 3D geometry or computer vision. The technique only requires the camera to observe a planar pattern shown at a few (at least two) different orientations. Either the camera or the planar pattern can be freely moved. The motion need not be known. Radial lens distortion is modeled. The proposed procedure consists of a closed-form solution, followed by a nonlinear refinement based on the maximum likelihood criterion. Both computer simulation and real data have been used to test the proposed technique, and very good results have been obtained. Compared with classical techniques which use expensive equipment such as two or three orthogonal planes, the proposed technique is easy to use and flexible. It advances 3D computer vision one step from laboratory environments to real world use. Index Terms Camera calibration, calibration from planes, 2D pattern, absolute conic, projective mapping, lens distortion, closed-form solution, maximum likelihood estimation, flexible setup. 1 Motivations Camera calibration is a necessary step in 3D computer vision in order to extract metric information from 2D images. Much work has been done, starting in the photogrammetry community (see [2, 4] to cite a few), and more recently in computer vision ([9, 8, 23, 7, 26, 24, 17, 6] to cite a few). We can classify those techniques roughly into two categories: photogrammetric calibration and selfcalibration. Photogrammetric calibration. Camera calibration is performed by observing a calibration object whose geometry in 3-D space is known with very good precision. Calibration can be done very efficiently [5]. The calibration object usually consists of two or three planes orthogonal to each other. Sometimes, a plane undergoing a precisely known translation is also used [23]. These approaches require an expensive calibration apparatus, and an elaborate setup. Self-calibration. Techniques in this category do not use any calibration object. Just by moving a camera in a static scene, the rigidity of the scene provides in general two constraints [17, 15] on the cameras internal parameters from one camera displacement by using image information alone. Therefore, if images are taken by the same camera with fixed internal parameters, correspondences between three images are sufficient to recover both the internal and external parameters which allow us to reconstruct 3-D structure up to a similarity [16, 13]. While this approach is very flexible, it is not yet mature [1]. Because there are many parameters to estimate, we cannot always obtain reliable results. Other techniques exist: vanishing points for orthogonal directions [3, 14], and calibration from pure rotation [11, 21]. Our current research is focused on a desktop vision system (DVS) since the potential for using DVSs is large. Cameras are becoming cheap and ubiquitous. A DVS aims at the general public, who are not experts in computer vision. A typical computer user will perform vision tasks only from time to time, so will not be willing to invest money for expensive equipment. Therefore, flexibility, robustness and low cost are important. The camera calibration technique described in this paper was developed with these considerations in mind. 2

4 The proposed technique only requires the camera to observe a planar pattern shown at a few (at least two) different orientations. The pattern can be printed on a laser printer and attached to a reasonable planar surface (e.g., a hard book cover). Either the camera or the planar pattern can be moved by hand. The motion need not be known. The proposed approach lies between the photogrammetric calibration and self-calibration, because we use 2D metric information rather than 3D or purely implicit one. Both computer simulation and real data have been used to test the proposed technique, and very good results have been obtained. Compared with classical techniques, the proposed technique is considerably more flexible. Compared with self-calibration, it gains considerable degree of robustness. We believe the new technique advances 3D computer vision one step from laboratory environments to the real world. Note that Bill Triggs [22] recently developed a self-calibration technique from at least 5 views of a planar scene. His technique is more flexible than ours, but has difficulty to initialize. Liebowitz and Zisserman [14] described a technique of metric rectification for perspective images of planes using metric information such as a known angle, two equal though unknown angles, and a known length ratio. They also mentioned that calibration of the internal camera parameters is possible provided at least three such rectified planes, although no experimental results were shown. The paper is organized as follows. Section 2 describes the basic constraints from observing a single plane. Section 3 describes the calibration procedure. We start with a closed-form solution, followed by nonlinear optimization. Radial lens distortion is also modeled. Section 4 studies configurations in which the proposed calibration technique fails. It is very easy to avoid such situations in practice. Section 5 provides the experimental results. Both computer simulation and real data are used to validate the proposed technique. In the Appendix, we provides a number of details, including the techniques for estimating the homography between the model plane and its image. 2 Basic Equations We examine the constraints on the camera s intrinsic parameters provided by observing a single plane. We start with the notation used in this paper. 2.1 Notation A 2D point is denoted by m = [u, v] T. A 3D point is denoted by M = [X, Y, Z] T. We use x to denote the augmented vector by adding 1 as the last element: m = [u, v, 1] T and M = [X, Y, Z, 1] T. A camera is modeled by the usual pinhole: the relationship between a 3D point M and its image projection m is given by s m = A [ R t ] M, (1) where s is an arbitrary scale factor, (R, t), called the extrinsic parameters, is the rotation and translation which relates the world coordinate system to the camera coordinate system, and A, called the camera intrinsic matrix, is given by α γ u A = β v 1 with (u, v ) the coordinates of the principal point, α and β the scale factors in image u and v axes, and γ the parameter describing the skewness of the two image axes. We use the abbreviation A T for (A 1 ) T or (A T ) 1. 3

5 2.2 Homography between the model plane and its image Without loss of generality, we assume the model plane is on Z = of the world coordinate system. Let s denote the i th column of the rotation matrix R by r i. From (1), we have X u s v = A [ r 1 r 2 r 3 t ] Y 1 1 = A [ r 1 r 2 t ] X Y. 1 By abuse of notation, we still use M to denote a point on the model plane, but M = [X, Y ] T since Z is always equal to. In turn, M = [X, Y, 1] T. Therefore, a model point M and its image m is related by a homography H: s m = H M with H = A [ r 1 r 2 t ]. (2) As is clear, the 3 3 matrix H is defined up to a scale factor. 2.3 Constraints on the intrinsic parameters Given an image of the model plane, an homography can be estimated (see Appendix A). Let s denote it by H = [ h 1 h 2 h 3 ]. From (2), we have [ h1 h 2 h 3 ] = λa [ r1 r 2 t ], where λ is an arbitrary scalar. Using the knowledge that r 1 and r 2 are orthonormal, we have h T 1 A T A 1 h 2 = (3) h T 1 A T A 1 h 1 = h T 2 A T A 1 h 2. (4) These are the two basic constraints on the intrinsic parameters, given one homography. Because a homography has 8 degrees of freedom and there are 6 extrinsic parameters (3 for rotation and 3 for translation), we can only obtain 2 constraints on the intrinsic parameters. Note that A T A 1 actually describes the image of the absolute conic [16]. In the next subsection, we will give an geometric interpretation. 2.4 Geometric Interpretation We are now relating (3) and (4) to the absolute conic. It is not difficult to verify that the model plane, under our convention, is described in the camera coordinate system by the following equation: [ ] T [ r3 xy ] r T 3 t z =, w where w = for points at infinity[ and ] w = [ 1 otherwise. ] This plane intersects the plane at infinity at r1 r2 a line, and we can easily see that and are two particular points on that line. Any point on it 4

6 is a linear combination of these two points, i.e., [ ] r1 x = a + b [ ] r2 = [ ] ar1 + br 2 Now, let s compute the intersection of the above line with the absolute conic. By definition, the point x, known as the circular point, satisfies: x T x =, i.e., (ar 1 + br 2 ) T (ar 1 + br 2 ) =, or a 2 + b 2 =. The solution is b = ±ai, where i 2 = 1. That is, the two intersection points are [ ] r1 ± ir x = a 2. Their projection in the image plane is then given, up to a scale factor, by m = A(r 1 ± ir 2 ) = h 1 ± ih 2. Point m is on the image of the absolute conic, described by A T A 1 [16]. This gives (h 1 ± ih 2 ) T A T A 1 (h 1 ± ih 2 ) =. Requiring that both real and imaginary parts be zero yields (3) and (4).. 3 Solving Camera Calibration This section provides the details how to effectively solve the camera calibration problem. We start with an analytical solution, followed by a nonlinear optimization technique based on the maximum likelihood criterion. Finally, we take into account lens distortion, giving both analytical and nonlinear solutions. 3.1 Closed-form solution Let B 11 B 12 B 13 B = A T A 1 B 12 B 22 B 23 B 13 B 23 B 33 = 1 α 2 γ α 2 β γ α 2 β Note that B is symmetric, defined by a 6D vector v γ u β α 2 β γ 2 α 2 β β 2 γ(v γ u β) α 2 β 2 v β 2 v γ u β α 2 β γ(v γ u β) α 2 β 2 v β 2 (v γ u β) 2 α 2 β 2 + v2 β (5) b = [B 11, B 12, B 22, B 13, B 23, B 33 ] T. (6) Let the i th column vector of H be h i = [h i1, h i2, h i3 ] T. Then, we have h T i Bh j = v T ijb (7) 5

7 with v ij = [h i1 h j1, h i1 h j2 + h i2 h j1, h i2 h j2, h i3 h j1 + h i1 h j3, h i3 h j2 + h i2 h j3, h i3 h j3 ] T. Therefore, the two fundamental constraints (3) and (4), from a given homography, can be rewritten as 2 homogeneous equations in b: [ v12 T ] (v 11 v 22 ) T b =. (8) If n images of the model plane are observed, by stacking n such equations as (8) we have Vb =, (9) where V is a 2n 6 matrix. If n 3, we will have in general a unique solution b defined up to a scale factor. If n = 2, we can impose the skewless constraint γ =, i.e., [, 1,,,, ]b =, which is added as an additional equation to (9). (If n = 1, we can only solve two camera intrinsic parameters, e.g., α and β, assuming u and v are known (e.g., at the image center) and γ =, and that is indeed what we did in [19] for head pose determination based on the fact that eyes and mouth are reasonably coplanar.) The solution to (9) is well known as the eigenvector of V T V associated with the smallest eigenvalue (equivalently, the right singular vector of V associated with the smallest singular value). Once b is estimated, we can compute all camera intrinsic matrix A. See Appendix B for the details. Once A is known, the extrinsic parameters for each image is readily computed. From (2), we have r 1 = λa 1 h 1 r 2 = λa 1 h 2 r 3 = r 1 r 2 t = λa 1 h 3 with λ = 1/ A 1 h 1 = 1/ A 1 h 2. Of course, because of noise in data, the so-computed matrix R = [r 1, r 2, r 3 ] does not in general satisfy the properties of a rotation matrix. Appendix C describes a method to estimate the best rotation matrix from a general 3 3 matrix. 3.2 Maximum likelihood estimation The above solution is obtained through minimizing an algebraic distance which is not physically meaningful. We can refine it through maximum likelihood inference. We are given n images of a model plane and there are m points on the model plane. Assume that the image points are corrupted by independent and identically distributed noise. The maximum likelihood estimate can be obtained by minimizing the following functional: n m m ij ˆm(A, R i, t i, M j ) 2, (1) i=1 j=1 where ˆm(A, R i, t i, M j ) is the projection of point M j in image i, according to equation (2). A rotation R is parameterized by a vector of 3 parameters, denoted by r, which is parallel to the rotation axis and whose magnitude is equal to the rotation angle. R and r are related by the Rodrigues formula [5]. Minimizing (1) is a nonlinear minimization problem, which is solved with the Levenberg-Marquardt Algorithm as implemented in Minpack [18]. It requires an initial guess of A, {R i, t i i = 1..n} which can be obtained using the technique described in the previous subsection. 6

8 3.3 Dealing with radial distortion Up to now, we have not considered lens distortion of a camera. However, a desktop camera usually exhibits significant lens distortion, especially radial distortion. In this section, we only consider the first two terms of radial distortion. The reader is referred to [2, 2, 4, 26] for more elaborated models. Based on the reports in the literature [2, 23, 25], it is likely that the distortion function is totally dominated by the radial components, and especially dominated by the first term. It has also been found that any more elaborated modeling not only would not help (negligible when compared with sensor quantization), but also would cause numerical instability [23, 25]. Let (u, v) be the ideal (nonobservable distortion-free) pixel image coordinates, and (ŭ, v) the corresponding real observed image coordinates. The ideal points are the projection of the model points according to the pinhole model. Similarly, (x, y) and ( x, y) are the ideal (distortion-free) and real (distorted) normalized image coordinates. We have [2, 25] x = x + x[k 1 (x 2 + y 2 ) + k 2 (x 2 + y 2 ) 2 ] y = y + y[k 1 (x 2 + y 2 ) + k 2 (x 2 + y 2 ) 2 ], where k 1 and k 2 are the coefficients of the radial distortion. The center of the radial distortion is the same as the principal point. From ŭ = u + α x + γ y and v = v + β y and assuming γ =, we have ŭ = u + (u u )[k 1 (x 2 + y 2 ) + k 2 (x 2 + y 2 ) 2 ] (11) v = v + (v v )[k 1 (x 2 + y 2 ) + k 2 (x 2 + y 2 ) 2 ]. (12) Estimating Radial Distortion by Alternation. As the radial distortion is expected to be small, one would expect to estimate the other five intrinsic parameters, using the technique described in Sect. 3.2, reasonable well by simply ignoring distortion. One strategy is then to estimate k 1 and k 2 after having estimated the other parameters, which will give us the ideal pixel coordinates (u, v). Then, from (11) and (12), we have two equations for each point in each image: [ (u u )(x 2 +y 2 ) (u u )(x 2 +y 2 ) 2 ] [ ] [ŭ u ] k1 (v v )(x 2 +y 2 ) (v v )(x 2 +y 2 ) 2 =. v v Given m points in n images, we can stack all equations together to obtain in total 2mn equations, or in matrix form as Dk = d, where k = [k 1, k 2 ] T. The linear least-squares solution is given by k 2 k = (D T D) 1 D T d. (13) Once k 1 and k 2 are estimated, one can refine the estimate of the other parameters by solving (1) with ˆm(A, R i, t i, M j ) replaced by (11) and (12). We can alternate these two procedures until convergence. Complete Maximum Likelihood Estimation. Experimentally, we found the convergence of the above alternation technique is slow. A natural extension to (1) is then to estimate the complete set of parameters by minimizing the following functional: n i=1 j=1 m m ij m(a, k 1, k 2, R i, t i, M j ) 2, (14) A typo was reported by Johannes Koester [[email protected]] via on Aug. 13, 28. 7

9 where m(a, k1, k2, R i, t i, M j ) is the projection of point M j in image i according to equation (2), followed by distortion according to (11) and (12). This is a nonlinear minimization problem, which is solved with the Levenberg-Marquardt Algorithm as implemented in Minpack [18]. A rotation is again parameterized by a 3-vector r, as in Sect An initial guess of A and {R i, t i i = 1..n} can be obtained using the technique described in Sect. 3.1 or in Sect An initial guess of k 1 and k 2 can be obtained with the technique described in the last paragraph, or simply by setting them to. 3.4 Summary The recommended calibration procedure is as follows: 1. Print a pattern and attach it to a planar surface; 2. Take a few images of the model plane under different orientations by moving either the plane or the camera; 3. Detect the feature points in the images; 4. Estimate the five intrinsic parameters and all the extrinsic parameters using the closed-form solution as described in Sect. 3.1; 5. Estimate the coefficients of the radial distortion by solving the linear least-squares (13); 6. Refine all parameters by minimizing (14). 4 Degenerate Configurations We study in this section configurations in which additional images do not provide more constraints on the camera intrinsic parameters. Because (3) and (4) are derived from the properties of the rotation matrix, if R 2 is not independent of R 1, then image 2 does not provide additional constraints. In particular, if a plane undergoes a pure translation, then R 2 = R 1 and image 2 is not helpful for camera calibration. In the following, we consider a more complex configuration. Proposition 1. If the model plane at the second position is parallel to its first position, then the second homography does not provide additional constraints. Proof. Under our convention, R 2 and R 1 are related by a rotation around z-axis. That is, cos θ sin θ R 1 sin θ cos θ = R 2, 1 where θ is the angle of the relative rotation. We will use superscript (1) and (2) to denote vectors related to image 1 and 2, respectively. It is clear that we have h (2) 1 = λ (2) (Ar (1) cos θ + Ar (2) sin θ) = λ(2) λ (1) (h(1) h (2) 2 = λ (2) ( Ar (1) sin θ + Ar (2) cos θ) = λ(2) λ Then, the first constraint (3) from image 2 becomes: h (2) T 1 A T A 1 h (2) 2 =λ(2) cos θ sin θ(h (1) 1 1 cos θ + h (1) 2 sin θ) (1) ( h(1) λ (1) [(cos2 θ sin 2 θ)(h (1) 1 T A T A 1 h (1) 1 h (1) 2 1 sin θ + h (1) 2 cos θ). T A T A 1 h (1) 2 ) T A T A 1 h (1) 2 )], 8

10 which is a linear combination of the two constraints provided by H 1. Similarly, we can show that the second constraint from image 2 is also a linear combination of the two constraints provided by H 1. Therefore, we do not gain any constraint from H 2. The result is self-evident because parallel planes intersect with the plane at infinity at the same circular points, and thus according to Sect. 2.4 they provide the same constraints. In practice, it is very easy to avoid the degenerate configuration: we only need to change the orientation of the model plane from one snapshot to another. Although the proposed technique will not work if the model plane undergoes pure translation, camera calibration is still possible if the translation is known. Please refer to Appendix D. 5 Experimental Results The proposed algorithm has been tested on both computer simulated data and real data. The closedform solution involves finding a singular value decomposition of a small 2n 6 matrix, where n is the number of images. The nonlinear refinement within the Levenberg-Marquardt algorithm takes 3 to 5 iterations to converge. 5.1 Computer Simulations The simulated camera has the following property: α = 125, β = 9, γ = (equivalent to ), u = 255, v = 255. The image resolution is The model plane is a checker pattern containing 1 14 = 14 corner points (so we usually have more data in the v direction than in the u direction). The size of pattern is 18cm 25cm. The orientation of the plane is represented by a 3D vector r, which is parallel to the rotation axis and whose magnitude is equal to the rotation angle. Its position is represented by a 3D vector t (unit in centimeters). Performance w.r.t. the noise level. In this experiment, we use three planes with r 1 = [2,, ] T, t 1 = [ 9, 12.5, 5] T, r 2 = [, 2, ] T, t 2 = [ 9, 12.5, 51] T, r 3 = 1 5 [ 3, 3, 15 ] T, t 3 = [ 1.5, 12.5, 525] T. Gaussian noise with mean and σ standard deviation is added to the projected image points. The estimated camera parameters are then compared with the ground truth. We measure the relative error for α and β, and absolute error for u and v. We vary the noise level from.1 pixels to 1.5 pixels. For each noise level, we perform 1 independent trials, and the results shown are the average. As we can see from Fig. 1, errors increase linearly with the noise level. (The error for γ is not shown, but has the same property.) For σ =.5 (which is larger than the normal noise in practical calibration), the errors in α and β are less than.3%, and the errors in u and v are around 1 pixel. The error in u is larger than that in v. The main reason is that there are less data in the u direction than in the v direction, as we said before. Performance w.r.t. the number of planes. This experiment investigates the performance with respect to the number of planes (more precisely, the number of images of the model plane). The orientation and position of the model plane for the first three images are the same as in the last subsection. From the fourth image, we first randomly choose a rotation axis in a uniform sphere, then apply a rotation angle of 3. We vary the number of images from 2 to 16. For each number, 1 trials of independent plane orientations (except for the first three) and independent noise with mean and 9

11 Relative error (%) alpha.2 beta Noise level (pixels) Absolute error (pixels) Noise level (pixels) u v Figure 1: Errors vs. the noise level of the image points Relative error (%) alpha beta Absolute error (pixels) u v number of planes number of planes Figure 2: Errors vs. the number of images of the model plane standard deviation.5 pixels are conducted. The average result is shown in Fig. 2. The errors decrease when more images are used. From 2 to 3, the errors decrease significantly. Performance w.r.t. the orientation of the model plane. This experiment examines the influence of the orientation of the model plane with respect to the image plane. Three images are used. The orientation of the plane is chosen as follows: the plane is initially parallel to the image plane; a rotation axis is randomly chosen from a uniform sphere; the plane is then rotated around that axis with angle θ. Gaussian noise with mean and standard deviation.5 pixels is added to the projected image points. We repeat this process 1 times and compute the average errors. The angle θ varies from 5 to 75, and the result is shown in Fig. 3. When θ = 5, 4% of the trials failed because the planes are almost parallel to each other (degenerate configuration), and the result shown has excluded those trials. Best performance seems to be achieved with an angle around 45. Note that in practice, when the angle increases, foreshortening makes the corner detection less precise, but this is not considered in this experiment. 5.2 Real Data The proposed technique is now routinely used in our vision group and also in the graphics group at Microsoft Research. Here, we provide the result with one example. The camera to be calibrated is an off-the-shelf PULNiX CCD camera with 6 mm lens. The image resolution is The model plane contains a pattern of 8 8 squares, so there are 256 corners. The size of the pattern is 17cm 17cm. It was printed with a high-quality printer and put on a glass. 1

12 Relative error (%) alpha beta Absolute error (pixels) u v Angle with the image plane (degrees) Angle with the image plane (degrees) Figure 3: Errors vs. the angle of the model plane w.r.t. the image plane Table 1: Results with real data of 2 through 5 images nb 2 images 3 images 4 images 5 images initial final σ initial final σ initial final σ initial final σ α β γ u v k k RMS Five images of the plane under different orientations were taken, as shown in Fig. 4. We can observe a significant lens distortion in the images. The corners were detected as the intersection of straight lines fitted to each square. We applied our calibration algorithm to the first 2, 3, 4 and all 5 images. The results are shown in Table 1. For each configuration, three columns are given. The first column (initial) is the estimation of the closed-form solution. The second column (final) is the maximum likelihood estimation (MLE), and the third column (σ) is the estimated standard deviation, representing the uncertainty of the final result. As is clear, the closed-form solution is reasonable, and the final estimates are very consistent with each other whether we use 2, 3, 4 or 5 images. We also note that the uncertainty of the final estimate decreases with the number of images. The last row of Table 1, indicated by RMS, displays the root of mean squared distances, in pixels, between detected image points and projected ones. The MLE improves considerably this measure. The careful reader may remark the inconsistency for k 1 and k 2 between the closed-form solution and the MLE. The reason is that for the closed-form solution, camera intrinsic parameters are estimated assuming no distortion, and the predicted outer points lie closer to the image center than the detected ones. The subsequent distortion estimation tries to spread the outer points and increase the scale in order to reduce the distances, although the distortion shape (with positive k 1, called pincushion distortion) does not correspond to the real distortion (with negative k 1, called barrel distortion). The nonlinear refinement (MLE) finally recovers the correct distortion shape. The estimated distortion parameters allow us to correct the distortion in the original images. Figure 5 displays the first two such distortion-corrected images, which should be compared with the first two images shown in Figure 4. We see clearly that the curved pattern in the original images is straightened. 11

13 relative errors (%) alpha beta absolute errors (pixels) u v noise level in model points (%) noise level in model points (%) relative error (% ) noise level in model points (%) k1 Figure 8: Sensitivity of camera calibration with respect to Gaussian noise in the model points Relative errors (% ) alpha beta aspect Systematic error in the model (%) absolute error (pixels) Systematic error in the model (%) u v Figure 9: Sensitivity of camera calibration with respect to systematic spherical non-planarity shown in Table 1) were calculated, and are depicted in Fig. 8. Obviously, all errors increase with the level of noise added to the model points. The pixel scale factors (α and β) remain very stable: the error is less than.2%. The coordinates of the principal point are quite stable: the errors are about 2 pixels for the noise level 15%. The estimated radial distortion coefficient k 1 becomes less useful, and the second term k 2 (not shown) is even less than k 1. In our current formulation, we assume that the exact position of the points in the model plane is known. If the model points are only known within certain precision, we can reformulate the problem, and we could expect smaller errors than reported here Systematic non-planarity of the model pattern In this section, we consider systematic non-planarity of the model pattern, e.g., when a printed pattern is attached to a soft book cover. We used the same configuration as in Sect The model plane 15

14 Relative errors (% ) alpha beta aspect Systematic error in the model (%) absolute errors (pixels) u 4 v Systematic error in the model (%) Figure 1: Sensitivity of camera calibration with respect to systematic cylindrical non-planarity was distorted in two systematic ways to simulate the non-planarity: spherical and cylindrical. With spherical distortion, points away from the center of the pattern are displaced in z according to z = p x 2 + y 2, where p indicates the non-planarity (the model points are coplanar when p = ). The displacement is symmetric around the center. With Cylindrical distortion, points are displaced in z according to z = px. Again, p indicates the non-planarity. This simulates bending of the model pattern around the vertical axis. Four images of the model pattern were used: the first is parallel to the image plane; the second is rotated from the first around the horizontal axis by 3 degrees; the third is rotated from the first around the vertical axis by 3 degrees; the fourth is rotated from the first around the diagonal axis by 3 degrees. Although model points are not coplanar, they were treated as coplanar, and the proposed calibration technique was applied. Gaussian noise with standard deviation.5 pixels was added to the image points, and 1 independent trials were conducted. The average calibration errors of the 1 trials are shown in Fig. 9 for spherical non-planarity and in Fig. 1 for cylindrical non-planarity. The horizontal axis indicates the increase in the non-planarity, which is measured as the ratio of the maximum z displacement to the size of the pattern. Therefore, 1% of non-planarity is equivalent to maximum 2.5cm of displacement in z, which does not likely happen in practice. Several observations can be made: Systematic non-planarity of the model has more effect on the calibration precision than random errors in the positions as described in the last subsection; Aspect ratio is very stable (.4% of error for 1% of non-planarity); Systematic cylindrical non-planarity is worse than systematic spherical non-planarity, especially for the coordinates of the principal point (u, v ). The reason is that cylindrical nonplanarity is only symmetric in one axis. That is also why the error in u is much larger than in v in our simulation; The result seems still usable in practice if there is only a few percents (say, less than 3%) of systematic non-planarity. The error in (u, v ) has been found by many researchers to have little effect in 3D reconstruction. As pointed out by Triggs in [22], the absolute error in (u, v ) is not geometrically meaningful. He proposes to measure the relative error with respect to the focal length, i.e., u /α and v /α. This is equivalent to measuring the angle between the true optical axis and the estimated one. Then, for 1% of cylindrical non-planarity (see Fig. 1), the relative error for u is 7.6%, comparable with those of α and β. 16

15 6 Conclusion In this paper, we have developed a flexible new technique to easily calibrate a camera. The technique only requires the camera to observe a planar pattern from a few (at least two) different orientations. We can move either the camera or the planar pattern. The motion does not need to be known. Radial lens distortion is modeled. The proposed procedure consists of a closed-form solution, followed by a nonlinear refinement based on maximum likelihood criterion. Both computer simulation and real data have been used to test the proposed technique, and very good results have been obtained. Compared with classical techniques which use expensive equipment such as two or three orthogonal planes, the proposed technique gains considerable flexibility. Acknowledgment Thanks go to Brian Guenter for his software of corner extraction and for many discussions, and to Bill Triggs for insightful comments. Thanks go to Andrew Zisserman for bringing his CVPR98 work [14] to my attention, which uses the same constraint but in different form, and for pointing out an error in my discussion on the case of pure translation. Thanks go to Bill Triggs and Gideon Stein for suggesting experiments described in Sect Thanks also go to the members of the Vision Group at MSR for encouragement and discussions. Anandan and Charles Loop have checked the English. A Estimation of the Homography Between the Model Plane and its Image There are many ways to estimate the homography between the model plane and its image. Here, we present a technique based on maximum likelihood criterion. Let M i and m i be the model and image points, respectively. Ideally, they should satisfy (2). In practice, they don t because of noise in the extracted image points. Let s assume that m i is corrupted by Gaussian noise with mean and covariance matrix Λ mi. Then, the maximum likelihood estimation of H is obtained by minimizing the following functional (m i ˆm i ) T Λ 1 m i (m i ˆm i ), i where ˆm i = 1 ] [ ht 1 M i h T 3 M i h T 2 M with h i, the i th row of H. i In practice, we simply assume Λ mi = σ 2 I for all i. This is reasonable if points are extracted independently with the same procedure. In this case, the above problem becomes a nonlinear least-squares one, i.e., min H i m i ˆm i 2. The nonlinear minimization is conducted with the Levenberg- Marquardt Algorithm as implemented in Minpack [18]. This requires an initial guess, which can be obtained as follows. Let x = [ h T 1, h T 2, h T 3 ]T. Then equation (2) can be rewritten as [ M T T u M T T M T v M T ] x =. When we are given n points, we have n above equations, which can be written in matrix equation as Lx =, where L is a 2n 9 matrix. As x is defined up to a scale factor, the solution is well known 17

16 to be the right singular vector of L associated with the smallest singular value (or equivalently, the eigenvector of L T L associated with the smallest eigenvalue). In L, some elements are constant 1, some are in pixels, some are in world coordinates, and some are multiplication of both. This makes L poorly conditioned numerically. Much better results can be obtained by performing a simple data normalization, such as the one proposed in [12], prior to running the above procedure. B Extraction of the Intrinsic Parameters from Matrix B Matrix B, as described in Sect. 3.1, is estimated up to a scale factor, i.e.,, B = λa T A with λ an arbitrary scale. Without difficulty, we can uniquely extract the intrinsic parameters from matrix B. v = (B 12 B 13 B 11 B 23 )/(B 11 B 22 B 2 12) λ = B 33 [B v (B 12 B 13 B 11 B 23 )]/B 11 α = λ/b 11 β = λb 11 /(B 11 B 22 B12 2 ) γ = B 12 α 2 β/λ u = γv /β B 13 α 2 /λ. C Approximating a 3 3 matrix by a Rotation Matrix The problem considered in this section is to solve the best rotation matrix R to approximate a given 3 3 matrix Q. Here, best is in the sense of the smallest Frobenius norm of the difference R Q. That is, we are solving the following problem: Since min R R Q 2 F subject to R T R = I. (15) R Q 2 F = trace((r Q) T (R Q)) = 3 + trace(q T Q) 2trace(R T Q), problem (15) is equivalent to the one of maximizing trace(r T Q). Let the singular value decomposition of Q be USV T, where S = diag (σ 1, σ 2, σ 3 ). If we define an orthogonal matrix Z by Z = V T R T U, then trace(r T Q) = trace(r T USV T ) = trace(v T R T US) 3 3 = trace(zs) = z ii σ i σ i. It is clear that the maximum is achieved by setting R = UV T because then Z = I. This gives the solution to (15). An excellent reference on matrix computations is the one by Golub and van Loan [1]. A typo was reported in formula u by Jiyong Ma [mailto:[email protected]] via an on April 18, 22. i=1 i=1 18

17 D Camera Calibration Under Known Pure Translation As said in Sect. 4, if the model plane undergoes a pure translation, the technique proposed in this paper will not work. However, camera calibration is possible if the translation is known like the setup in Tsai s technique [23]. From (2), we have t = αa 1 h 3, where α = 1/ A 1 h 1. The translation between two positions i and j is then given by t (ij) = t (i) t (j) = A 1 (α (i) h (i) 3 α(j) h (j) 3 ). (Note that although both H (i) and H (j) are estimated up to their own scale factors, they can be rescaled up to a single common scale factor using the fact that it is a pure translation.) If only the translation direction is known, we get two constraints on A. If we know additionally the translation magnitude, then we have another constraint on A. Full calibration is then possible from two planes. 19

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